# combination equation

• Jun 18th 2009, 12:21 PM
dhiab
combination equation
• Jun 18th 2009, 01:21 PM
TheAbstractionist
Observe that

$\displaystyle ^{n-1}\mathrm C_{p-1}\,+\,^{n-1}\mathrm C_p\ =\ ^n\mathrm C_p$

Hence

$\displaystyle x^2\,-\,^n\mathrm C_px\,+\,^{n-1}\mathrm C_{p-1}\cdot^{n-1}\mathrm C_p$

$\displaystyle =\ x^2\,-\,\left(^{n-1}\mathrm C_{p-1}\,+\,^{n-1}\mathrm C_p\right)x\,+\,^{n-1}\mathrm C_{p-1}\cdot^{n-1}\mathrm C_p$

$\displaystyle =\ \left(x\,-\,^{n-1}\mathrm C_{p-1}\right)\left(x\,-\,^{n-1}\mathrm C_p\right)$
• Jun 18th 2009, 10:30 PM
dhiab
Quote:

Originally Posted by TheAbstractionist
Observe that
$\displaystyle ^{n-1}\mathrm C_{p-1}\,+\,^{n-1}\mathrm C_p\ =\ ^n\mathrm C_p$
Hence
$\displaystyle x^2\,-\,^n\mathrm C_px\,+\,^{n-1}\mathrm C_{p-1}\cdot^{n-1}\mathrm C_p$
$\displaystyle =\ x^2\,-\,\left(^{n-1}\mathrm C_{p-1}\,+\,^{n-1}\mathrm C_p\right)x\,+\,^{n-1}\mathrm C_{p-1}\cdot^{n-1}\mathrm C_p$

$\displaystyle =\ \left(x\,-\,^{n-1}\mathrm C_{p-1}\right)\left(x\,-\,^{n-1}\mathrm C_p\right)$

Hello : Thank you
it is necessary to discuss the number of resolution according to paramétres n , p (Itwasntme)